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Properties

Label 2-468-13.4-c3-0-9
Degree $2$
Conductor $468$
Sign $0.969 - 0.246i$
Analytic cond. $27.6128$
Root an. cond. $5.25479$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.9i·5-s + (26.4 − 15.2i)7-s + (36.8 + 21.2i)11-s + (33.9 − 32.2i)13-s + (−26.3 − 45.6i)17-s + (−111. + 64.5i)19-s + (54.2 − 93.9i)23-s − 18.5·25-s + (44.1 − 76.5i)29-s − 32.1i·31-s + (183. + 317. i)35-s + (300. + 173. i)37-s + (−157. − 90.9i)41-s + (37.8 + 65.5i)43-s + 307. i·47-s + ⋯
L(s)  = 1  + 1.07i·5-s + (1.42 − 0.824i)7-s + (1.00 + 0.582i)11-s + (0.724 − 0.688i)13-s + (−0.375 − 0.650i)17-s + (−1.34 + 0.778i)19-s + (0.491 − 0.852i)23-s − 0.148·25-s + (0.282 − 0.489i)29-s − 0.186i·31-s + (0.884 + 1.53i)35-s + (1.33 + 0.769i)37-s + (−0.600 − 0.346i)41-s + (0.134 + 0.232i)43-s + 0.953i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(27.6128\)
Root analytic conductor: \(5.25479\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :3/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.505731380\)
\(L(\frac12)\) \(\approx\) \(2.505731380\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-33.9 + 32.2i)T \)
good5 \( 1 - 11.9iT - 125T^{2} \)
7 \( 1 + (-26.4 + 15.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-36.8 - 21.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (26.3 + 45.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (111. - 64.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-54.2 + 93.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-44.1 + 76.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 32.1iT - 2.97e4T^{2} \)
37 \( 1 + (-300. - 173. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (157. + 90.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-37.8 - 65.5i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 307. iT - 1.03e5T^{2} \)
53 \( 1 - 693.T + 1.48e5T^{2} \)
59 \( 1 + (314. - 181. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-214. - 371. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (169. + 98.1i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-907. + 523. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 234. iT - 3.89e5T^{2} \)
79 \( 1 - 409.T + 4.93e5T^{2} \)
83 \( 1 + 13.1iT - 5.71e5T^{2} \)
89 \( 1 + (552. + 319. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.06e3 - 612. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.79057557818388410270027929272, −10.03024078404173314466350198287, −8.692806844116309086830722730783, −7.87757185205050264806694714304, −6.95985228656816957434545721159, −6.19073272634838074832069861345, −4.67455306284264714812859967926, −3.91174172212997904210333341157, −2.43153494763520883815560188095, −1.09562289194036741743452991968, 1.09633555607508124195177276802, 2.04873330578836904690069129941, 3.96203553646849543080974516559, 4.79785845979626492873680270905, 5.74454869380252679624809953676, 6.79614812460492676369548097324, 8.351458109010673190235690849874, 8.645246381645969914649869621246, 9.288635436461500075071976123630, 10.95457493904372565103560053842

Graph of the $Z$-function along the critical line